A308524 Number of essentially 3-connected rooted toroidal maps with n edges.

0, 0, 1, 2, 11, 40, 166, 658, 2647, 10592, 42446, 169972, 680670, 2725320, 10910992, 43678882, 174843151, 699839680, 2801078662, 11210671612, 44866276906, 179552951440, 718539964132, 2875389341332, 11506176209206, 46042099714240, 184234059839116, 737184620655368

Offset: 0

Author

Nicolas Bonichon, Jun 05 2019

Keywords

• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1000
• Nicolas Bonichon, Éric Fusy, Benjamin Lévêque, A bijection for essentially 3-connected toroidal maps, arXiv:1907.04016 [math.CO], 2019.

Crossrefs

Cf. A308523, A308526, A289208, A006422.

Programs

Maple

dev_A := 0; n := 20; dev_A := series(RootOf(A-x*(1+A)^2, A), x = 0, n+1);
seq(coeff(series(subs(A = dev_A, A^2*(1+A)/((1+2*A)*(1-A)^2*(1+3*A))), x, n+1), x, k), k = 0 .. n);
# second Maple program:
a:= proc(n) option remember; `if`(n<6, [0, 0, 1, 2, 11, 40][n+1],
     ((37*n^2-258*n+401)*a(n-1)-6*(2*n^2-25*n+88)*a(n-2)
      -48*(3*n^2-23*n+45)*a(n-3)-32*(n-4)*(2*n-7)*a(n-4))
      /((6*(n-1))*(n-5)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 07 2019

Mathematica

CoefficientList[Series[x*(1 + 8*x + (2*x - 1)*Sqrt[1 - 4*x])/(2*(2 + x)*(1 - 4*x)*(3 + 4*x)), {x, 0, 30}], x] (* Vaclav Kotesovec, Jun 25 2019 *)

Formula

G.f.: A^2*(1+A)/((1+2*A)*(1-A)^2*(1+3*A)) where A=x*(1+A)^2.

G.f.: x*(1 + 8*x + (2*x - 1)*sqrt(1 - 4*x))/(2*(2 + x)*(1 - 4*x)*(3 + 4*x)). - Vaclav Kotesovec, Jun 25 2019

a(n) ~ 2^(2*n - 3) / 3. - Vaclav Kotesovec, Jun 25 2019